Leakage diagnosis method and system for water supply network
By combining Fourier transform and frequency domain analysis with frequency region division and sliding window technology, the accuracy problem of leak signal detection in water supply networks was solved, enabling precise location of leak signals and optimization of signal processing, thereby improving the safety and reliability of the water supply system.
Patent Information
- Application Number
- CN202511640425.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-11-11
- Publication Date
- 2025-12-12
- Estimated Expiration
- 2045-11-11
AI Technical Summary
Existing technologies have limitations in processing leakage signals from water supply networks. Fourier transform is insufficient to accurately capture the time-varying characteristics of leakage signals, thus affecting the accuracy of leakage detection.
The signal is converted from the time domain to the frequency domain by Fourier transform, divided into low, medium and high frequency regions, and the frequency difference is calculated. Sliding window division and frequency domain similarity calculation are used, and multiple sensors are combined to locate the leak location, thereby improving detection sensitivity and accuracy.
It enables precise location of leakage signals and optimized signal processing, improving the safety and reliability of the water supply system and enhancing the accuracy and sensitivity of leakage detection.
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Figure CN121117643A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of data processing. More specifically, this invention relates to a method and system for diagnosing leaks in water supply networks. Background Technology
[0002] As a vital component of a city's lifeline, the safe operation of water supply networks is a fundamental guarantee for urban construction and people's lives. However, due to various factors, water supply network leaks occur frequently, causing considerable inconvenience to urban management and residents. With the development of technology, modern water supply network monitoring systems utilize next-generation information technologies such as artificial intelligence of things (AI), the Internet of Things (IoT), big data, mobile internet, industrial internet, and hydraulic models to monitor the operational status and water quality of the water supply network in real time. These technologies allow for the timely detection and location of leaks, enabling appropriate repair measures to be taken. Furthermore, by analyzing historical data, the monitoring system can predict future leakage risks and issue early warnings, providing strong protection for the city's water supply safety.
[0003] Currently, existing technologies, such as the patent application with publication number CN117823830A, disclose a method and system for leak detection using an intelligent detection ball for water supply pipelines. This leak detection method includes: setting an ultrasonic transmitter in the pipeline section to be inspected and inserting the detection ball into the pipeline to collect acoustic information; decomposing the collected acoustic signal into multiple decomposition signals according to different decomposition parameters, obtaining the optimal decomposition parameters and the corresponding optimal decomposition signal; converting the signal from the time domain to the frequency domain using Fourier transform to obtain the ultrasonic characteristic frequency domain components of the optimal decomposition signal; identifying the ultrasonic characteristic frequency domain components using a pattern recognition method to obtain the signal corresponding to the pipeline leak and the time of the leak; and using the time of the leak, combined with the correspondence between location and time, determining the specific location of the leak, thereby achieving leak detection.
[0004] However, Fourier transform has certain limitations when processing non-stationary signals such as leakage signals. For example, the frequency components of water supply pipeline leakage signals change over time, and there is signal noise inside the pipeline, making it difficult to accurately capture the time-varying characteristics of leakage signals, thus affecting the accuracy of leakage detection. Summary of the Invention
[0005] To address the limitations of processing non-stationary signals like leakage signals, which makes it difficult to accurately capture the time-varying characteristics of leakage signals and thus affect the accuracy of leakage detection, this invention provides solutions in the following aspects.
[0006] In the first aspect, a leak diagnosis method for a water supply network includes: Collect historical pre-processed normal and leakage signals from the water supply pipeline; The historical normal signal and leakage signal are transformed from the time domain to the frequency domain by Fourier transform. The signal is divided into low, medium and high frequency regions according to the frequency coverage. In each region, the frequency difference between the historical normal signal and leakage signal is calculated to determine the target frequency of each region. The collected historical normal signals and leaked signals are divided into sliding windows according to the target frequency of each region. The frequency domain similarity between the real-time signal and the historical normal signals and leaked signals at each time point in each region window is calculated. Based on the frequency domain similarity, the probability that the real-time signal at that time is a leaked signal is calculated. If the probability is greater than a preset threshold, it is determined to be a leaked signal. The location of the leak is determined by the moment when multiple sensors first pick up the leak signal, wherein the moment when the sensors first pick up the leak signal is obtained by calculating the probability that the real-time signal is a leak signal is greater than a preset threshold.
[0007] This invention transforms the signal from the time domain to the frequency domain using Fourier transform and divides it into low, medium, and high frequency regions. This allows for a clearer identification of the frequency differences between normal and leak signals. A target frequency is determined within each frequency region, further enhancing the ability to capture leak signal characteristics. Then, through sliding window division and frequency domain similarity calculation, signal changes can be monitored in real time, and features similar to leak signals can be quickly identified in the frequency domain. When the frequency domain similarity of the real-time signal exceeds a preset threshold at a certain moment, it can be determined as a leak signal, thus more accurately determining whether the signal is leaking. Finally, by utilizing the moment when multiple sensors first pick up the leak signal, the leak location can be located using methods such as time difference calculation. This improves the sensitivity and accuracy of leak detection, achieves precise positioning and optimizes the signal processing flow, and helps improve the safety and reliability of the entire water supply system.
[0008] Preferably, the frequency difference between the historical normal signal and the leakage signal satisfies the following relationship: In the formula, For historical normal signals and leakage signals at frequency Frequency domain differences The number of normal signals in history. The number of historical leak signals, Indicates the first A historical normal signal at frequency The spectral amplitude below, Indicates the first A historical leakage signal at frequency The spectral amplitude below.
[0009] By comparing different frequencies The value can reveal which frequency components of the leaked signal differ most from the normal signal, thus allowing for the extraction of representative frequency domain features.
[0010] Preferably, determining the target frequency for each region includes: The frequency corresponding to the maximum frequency domain difference in each region is taken as the target frequency for each region.
[0011] By selecting the frequency corresponding to the maximum frequency domain difference as the target frequency, the most distinctive frequency components in the signal can be accurately located. These frequency components often contain the core information of signal state changes (such as normal and fault states).
[0012] Preferably, determining the target frequency for each region further includes: For each region, calculate the relative change between the frequency domain difference of the historical normal signal and leakage signal at the target frequency and the mean of the frequency domain difference in the region where the target frequency is located, and label it as the degree of frequency domain difference between the historical normal signal and leakage signal at the target frequency. The optimal partitioning result is the region with the largest ratio between the frequency domain difference between the historical normal signal and the leakage signal at the target frequency in the low-frequency region and the frequency domain difference between the historical normal signal and the leakage signal at the target frequency in the high-frequency region. The target frequency of each region in the optimal partitioning result is then taken as the optimal target frequency.
[0013] By comparing the ratio of the frequency domain differences between historical normal signals and leakage signals at the target frequency in the low-frequency and high-frequency regions, the region division with the largest ratio is selected as the optimal division. This means that when selecting regions, priority is given to those frequency bands where the frequency domain differences between normal signals and leakage signals are most significant and easiest to distinguish. Based on the optimal division results, the target frequency of each region is determined as the optimal target frequency, so that in practical applications, these frequencies can be monitored and analyzed more specifically, improving the accuracy and efficiency of signal processing.
[0014] Preferably, the frequency domain similarity between the real-time signal and the historical leaked signal within the low-frequency region window at each moment satisfies the following relationship: In the formula, The frequency domain similarity between the real-time signal and the historical leakage signal within a window divided based on the target frequency in the low-frequency region is calculated. For real-time signals in the low-frequency region Frequency within a window The spectral amplitude below, Historical leak signal In the low frequency region Frequency within a window The spectral amplitude below, This indicates the number of historical leakage signals. This represents the total number of windows divided based on the target frequency in the low-frequency region. The exponential function is represented; the frequency domain similarity in the mid-frequency region window and the high-frequency region window is calculated in the same way as the frequency domain similarity in the low-frequency region window.
[0015] By calculating the frequency domain similarity between the real-time signal and the historical leakage signal, it can be determined whether the real-time signal matches some known historical leakage signals.
[0016] Preferably, the probability that the real-time signal is a leaky signal satisfies the following relationship: In the formula, The possibility that the real-time signal is a leaked signal. The normalized weights For the first Frequency similarity between real-time signals and historical leaked signals within a frequency region For the first Frequency similarity between real-time signals and historical normal signals within a frequency region.
[0017] By comparing the frequency similarity of real-time signals with historical leak signals and normal signals, it is possible to more accurately determine whether a leak exists. Furthermore, by considering information from multiple frequency regions, the impact of noise or anomalies in a single frequency region on the judgment can be reduced, thereby improving the robustness of the system.
[0018] Preferably, the normalized weights satisfy the following relationship: In the formula, The normalized weights In the first The degree of frequency domain difference between normal and leaked signals at their target frequency within a frequency region.
[0019] In a second aspect, a leak diagnosis system for a water supply network includes a processor and a memory, wherein the memory stores computer program instructions that, when executed by the processor, implement any of the leak diagnosis methods for a water supply network described above.
[0020] The beneficial effects of this invention are: This invention effectively addresses the problem of processing non-stationary signals such as leakage signals through frequency domain analysis, frequency region division, sliding window technology, quantization judgment, and location positioning. By focusing on the frequency characteristics that best distinguish normal signals from leakage signals and capturing the time-varying characteristics of the signal in real time, it improves the accuracy of leakage detection. Attached Figure Description
[0021] The above and other objects, features, and advantages of exemplary embodiments of the present invention will become readily apparent upon reading the following detailed description with reference to the accompanying drawings. In the drawings, several embodiments of the invention are illustrated by way of example and not limitation, and like or corresponding reference numerals denote like or corresponding parts, wherein: Figure 1 This is a flowchart of steps S1-S4 in a method for diagnosing leaks in a water supply network according to an embodiment of the present invention. Detailed Implementation
[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0023] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.
[0024] Reference Figure 1 A method for diagnosing leaks in a water supply network includes steps S1-S4, as detailed below: S1: Collects normal and leakage signals from the water supply pipeline after historical pre-processing.
[0025] Specifically, acoustic sensors are installed at the beginning and end of each section of the water supply network to collect historical normal and leakage signals in the pipeline, as well as real-time signals of the water supply network to be diagnosed.
[0026] A high-pass filter is then used to remove environmental noise such as fluid flow and mechanical noise, resulting in historical normal and leakage signals from the pre-processed water supply network.
[0027] The sensor's sampling frequency was set to 10kHz, and the acquisition time was set to 10 minutes.
[0028] S2: The historical normal signal and leakage signal are transformed from the time domain to the frequency domain through Fourier transform. The signal is divided into low, medium and high frequency regions according to the frequency coverage. In each region, the frequency difference between the historical normal signal and leakage signal is calculated to determine the target frequency of each region.
[0029] In water supply pipeline systems, leakage signals are often quite weak, while the noise signals from normal fluid flow are much stronger. In the time domain, it is difficult to directly identify leaks simply by observing the amplitude, slope, and other characteristics of the signal. Therefore, Fourier transform is used to convert the signal to the frequency domain for analysis.
[0030] Specifically, Fourier transform is used to convert both the historical normal signal and the leakage signal acquired in S1 into frequency domain signals. Since the leakage signal usually exhibits an enhancement of high-frequency components or abrupt changes in characteristic frequency bands in the frequency domain, while environmental noise is usually concentrated in low frequencies and the low-frequency components attenuate less during signal propagation, the signal can be initially divided into three frequency regions: low, medium, and high, based on the frequency coverage. For example, one division result is given: the low-frequency region is (30 Hz, 300 kHz), the medium-frequency region is (300 kHz, 3 MHz), and the high-frequency region is (3 MHz, 30 MHz). Various other division results can be given in other embodiments.
[0031] Furthermore, in order to find the significant differences between historical leakage signals and normal signals in the frequency domain, the frequency domain differences between historical normal signals and leakage signals at each frequency are calculated by comparing the spectral amplitudes of normal signals and leakage signals at the same frequency within each frequency domain region.
[0032] For example, any frequency within the low, medium, and high frequency domains can be selected. Calculate historical normal signals and leakage signals at frequencies The frequency domain difference, i.e., satisfying the following relationship:
[0033] In the formula, For historical normal signals and leakage signals at frequency Frequency domain differences The number of normal signals in history. The number of historical leak signals, Indicates the first A historical normal signal at frequency The spectral amplitude below, Indicates the first A historical leakage signal at frequency The spectral amplitude below.
[0034] By calculating at frequency The average difference in spectral amplitude between all historical normal signals and all historical leakage signals is used. The greater the difference, the more obvious the characteristics of the leakage signal at that frequency.
[0035] Based on the above historical normal signals and leakage signals at frequency The frequency domain difference can be calculated similarly for all other frequencies, including historical normal signals and leakage signals.
[0036] Furthermore, for the low, medium, and high frequency regions, each region will be further divided into... The frequency corresponding to the maximum value is marked as the target frequency. This means that within each region, the difference between the normal signal and the leakage signal at the target frequency is the most significant.
[0037] Furthermore, for a target frequency within a frequency range, the relative differences between historical normal signals and leakage signals at the target frequency are calculated, satisfying the following relationship:
[0038] In the formula, To be at the target frequency The degree of frequency domain difference between the normal signal and the leaked signal. For historical normal signals and leakage signals at the target frequency Frequency domain differences For historical normal signals and leakage signals at the target frequency The mean of the frequency domain differences across all frequencies within the region.
[0039] The above The larger the value, the higher the target frequency. The more significant the frequency domain difference between normal signals and leaked signals within the target frequency, the better.
[0040] Based on the degree of frequency domain difference between different frequency regions at the target frequency, the three frequency regions of low, medium and high frequency in the above preliminary division results are evaluated in order to find a region division scheme that can maximize the difference between the leaked signal and the normal signal.
[0041] To evaluate the initial division of the three frequency regions (low, medium, and high) mentioned above, the degree of frequency domain difference between the different regions at the target frequency is considered. In particular, attention is paid to the degree of frequency domain difference between the low-frequency and high-frequency regions at the target frequency. The optimal division result is the region with the largest ratio of the degree of frequency domain difference between the historical normal signal and leakage signal at the target frequency in the low-frequency region to that in the high-frequency region. This means that under this division, the signal difference between the low-frequency and high-frequency regions is most significant, which may be more beneficial for subsequent signal processing or analysis.
[0042] Then, based on the calculation steps of the relevant frequency domain differences in the preliminary division results, the optimal target frequency of each frequency domain range in the optimal division results is found.
[0043] S3: Divide the collected historical normal signals and leakage signals into sliding windows according to the target frequency of each region, calculate the frequency domain similarity between the real-time signal and the historical normal signals and leakage signals in each region window at each moment, and calculate the probability that the real-time signal at that moment is a leakage signal based on the frequency domain similarity. If the probability is greater than a preset threshold, it is determined to be a leakage signal.
[0044] First, the historical normal signals and leakage signals collected by S1 are divided into sliding windows according to the target frequencies of the low, medium, and high frequency regions in the optimal partitioning result. The length of each window is the reciprocal of the corresponding target frequency, and the number of windows depends on the length of the collected signals. Then, for each moment of the real-time signal, the frequency domain similarity between the real-time signal within the window partitioned based on the target frequencies of the low, medium, and high frequency regions and all historical leakage signals is calculated. The general steps are as follows: For each historical leakage signal, calculate its difference with the real-time signal in the frequency domain, i.e., the absolute value of the difference in spectral amplitude. For each window, calculate the average of the absolute values of the differences in spectral amplitude corresponding to all frequency points. For all windows, calculate the exponential function of the average of the absolute values of the differences in spectral amplitude corresponding to all frequency points. This yields the frequency domain similarity between the real-time signal and all historical leakage signals within the window divided by the target frequency based on the low, medium, and high frequency regions at the corresponding time.
[0045] Taking the low-frequency region as an example, for each historical leakage signal (common (number), which is slidably divided into multiple windows (total) (number), for each window Calculate the sum of the absolute values of the spectral amplitude differences between the real-time signal and the historical leakage signal at all frequencies within the window; for each historical leakage signal... The sum of the absolute values of the differences calculated above is divided by the total number of windows corresponding to the historical leakage signals to obtain the average difference of each historical leakage signal. The average of all historical signals is summed and then divided by the number of historical leakage signals to obtain the overall average difference. The overall average difference is negative and then used as the exponent of the exponential function to calculate the frequency domain similarity between the real-time signal and the historical leakage signal within the window based on the target frequency division of the low-frequency region.
[0046] For each moment of the real-time signal, the frequency domain similarity between the real-time signal and the historical leaked signal within the window divided based on the target frequency of the low-frequency region satisfies the following relationship:
[0047] In the formula, The frequency domain similarity between the real-time signal and the historical leakage signal within a window divided based on the target frequency in the low-frequency region is calculated. For real-time signals in the low-frequency region Frequency within a window The spectral amplitude below, Historical leak signal In the low frequency region Frequency within a window The spectral amplitude below, This indicates the number of historical leakage signals. This represents the total number of windows divided based on the target frequency in the low-frequency region. This represents an exponential function.
[0048] The calculation methods for frequency domain similarity within the mid-frequency region window and the high-frequency region window are the same as those for frequency domain similarity within the low-frequency region window, and will not be elaborated further here.
[0049] Similarly, for each moment of the real-time signal, the calculation methods for the frequency domain similarity between the real-time signal and the historical normal signal within the window divided by the target frequency of the low-frequency region and the frequency domain similarity between the real-time signal and the historical leaked signal within the window divided by the target frequency of the low-frequency region are the same, and will not be elaborated on here.
[0050] Furthermore, the probability that the real-time signal at each moment is a leaked signal is calculated based on the frequency similarity corresponding to the low, medium, and high frequency regions. Taking any moment of the real-time signal as an example, the following relationship is satisfied:
[0051] In the formula, The possibility that the real-time signal is a leaked signal. The normalized weights For the first Frequency similarity between real-time signals and historical leaked signals within a frequency region For the first Frequency similarity between real-time signals and historical normal signals within a frequency region.
[0052] Among them, the normalized weights The relation is satisfied as follows:
[0053] In the formula, In the first The degree of frequency domain difference between normal and leaked signals at their target frequency within a frequency region.
[0054] Based on the above calculations Similarly, the probability that the real-time signal is a leaking signal at all times can be calculated. Then, the real-time signal at the corresponding time with a probability greater than 1 is marked as a leaking signal, and the real-time signal at the corresponding time with a probability less than or equal to 1 is marked as a normal signal.
[0055] S4: Locate the leak location based on the time when multiple sensors first pick up the leak signal, wherein the time when the sensor first picks up the leak signal is obtained when the probability that the calculated real-time signal is a leak signal is greater than a preset threshold.
[0056] Specifically, assume there are three sensors located at coordinates (x1, y1), (x2, y2), and (x3, y3). When a leak occurs, the leak signal propagates along the pipe at a certain speed v and is captured by the three sensors at different times. Let t1, t2, and t3 be the times when the three sensors first capture the leak signal. According to the principle of leak signal propagation, the relationship between the distance from the leak location (x, y) to the first sensor and the signal propagation speed and time is expressed as: The relationship between the distance from the leak location (x, y) to the second and third sensors and the signal propagation speed and time is consistent with the expression for the first sensor mentioned above.
[0057] By solving the three relationships above, the exact location (x, y) of the leak can be found.
[0058] The system includes a processor and a memory, the memory storing computer program instructions that, when executed by the processor, implement the leak diagnosis method for a water supply network according to the first aspect of the present invention.
[0059] The system also includes other components well known to those skilled in the art, such as communication buses and communication interfaces, the settings and functions of which are known in the art and will not be described in detail here.
[0060] In this invention, the aforementioned memory can be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, apparatus, or device. For example, a computer-readable storage medium can be any suitable magnetic or magneto-optical storage medium, such as Resistive Random Access Memory (RRAM), Dynamic Random Access Memory (DRAM), Static Random Access Memory (SRAM), Enhanced Dynamic Random Access Memory (EDRAM), High-Bandwidth Memory (HBM), Hybrid Memory Cube (HMC), etc., or any other medium that can be used to store desired information and can be accessed by an application, module, or both. Any such computer storage medium can be part of a device or accessible to or connected to a device.
[0061] In the description of this specification, "multiple" or "several" means at least two, such as two, three or more, unless otherwise explicitly specified.
[0062] While this specification has shown and described numerous embodiments of the invention, it will be apparent to those skilled in the art that such embodiments are provided by way of example only. Many modifications, alterations, and alternatives will occur to those skilled in the art without departing from the spirit and essence of the invention. It should be understood that various alternatives to the embodiments of the invention described herein may be employed in the practice of this invention.
Claims
1. A method for diagnosing leaks in water supply networks, characterized in that, include: Collect historical pre-processed normal and leakage signals from the water supply pipeline; The historical normal signal and leakage signal are transformed from the time domain to the frequency domain by Fourier transform. The signal is divided into low, medium and high frequency regions according to the frequency coverage. In each region, the frequency difference between the historical normal signal and leakage signal is calculated to determine the target frequency of each region. The collected historical normal signals and leaked signals are divided into sliding windows according to the target frequency of each region. The frequency domain similarity between the real-time signal and the historical normal signals and leaked signals at each time point in each region window is calculated. Based on the frequency domain similarity, the probability that the real-time signal at that time is a leaked signal is calculated. If the probability is greater than a preset threshold, it is determined to be a leaked signal. The location of the leak is determined by the moment when multiple sensors first pick up the leak signal, wherein the moment when the sensors first pick up the leak signal is obtained by calculating the probability that the real-time signal is a leak signal is greater than a preset threshold.
2. The method for leak diagnosis in a water supply network according to claim 1, characterized in that, The frequency difference between the historical normal signal and the leaked signal satisfies the following relationship: In the formula, For historical normal signals and leakage signals at frequency Frequency domain differences The number of normal signals in history. The number of historical leak signals, Indicates the first A historical normal signal at frequency The spectral amplitude below, Indicates the first A historical leakage signal at frequency The spectral amplitude below.
3. The method for leak diagnosis in a water supply network according to claim 2, characterized in that, Determining the target frequency for each region includes: The frequency corresponding to the maximum frequency domain difference in each region is taken as the target frequency for each region.
4. The method for leak diagnosis in a water supply network according to claim 3, characterized in that, The determination of the target frequency for each region also includes: For each region, calculate the relative change between the frequency domain difference of the historical normal signal and leakage signal at the target frequency and the mean of the frequency domain difference in the region where the target frequency is located, and label it as the degree of frequency domain difference between the historical normal signal and leakage signal at the target frequency. The optimal partitioning result is the region with the largest ratio between the frequency domain difference between the historical normal signal and the leakage signal at the target frequency in the low-frequency region and the frequency domain difference between the historical normal signal and the leakage signal at the target frequency in the high-frequency region. The target frequency of each region in the optimal partitioning result is then taken as the optimal target frequency.
5. A method for diagnosing leaks in a water supply network according to claim 4, characterized in that, The frequency domain similarity between the real-time signal and the historical leaked signal within the low-frequency region window at each moment satisfies the following relationship: In the formula, The frequency domain similarity between the real-time signal and the historical leakage signal within a window divided based on the target frequency in the low-frequency region is calculated. For real-time signals in the low-frequency region Frequency within a window The spectral amplitude below, Historical leak signal In the low frequency region Frequency within a window The spectral amplitude below, This indicates the number of historical leakage signals. This represents the total number of windows divided based on the target frequency in the low-frequency region. The exponential function is represented; the frequency domain similarity in the mid-frequency region window and the high-frequency region window is calculated in the same way as the frequency domain similarity in the low-frequency region window.
6. A method for diagnosing leaks in a water supply network according to claim 5, characterized in that, The probability that the real-time signal is a leaking signal satisfies the following relationship: In the formula, The possibility that the real-time signal is a leaked signal. The normalized weights For the first Frequency similarity between real-time signals and historical leaked signals within a frequency region For the first Frequency similarity between real-time signals and historical normal signals within a frequency region.
7. A method for diagnosing leaks in a water supply network according to claim 6, characterized in that, The normalized weights satisfy the following relation: In the formula, The normalized weights In the first The degree of frequency domain difference between normal and leaked signals at their target frequency within a frequency region.
8. A leak diagnosis system for water supply networks, characterized in that, include: A processor and a memory, the memory storing computer program instructions that, when executed by the processor, implement the leak diagnosis method for a water supply network according to any one of claims 1-7.
Citation Information
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